Question

Let $(\lambda ,2,1)$ be a point on the plane which passes through the point $(4,–2,2).$ If the plane is perpendicular to the line joining the points $(–2,–21,29)$ and $(–1,–16,23),$ then ${\left(\frac{\lambda }{11}\right)}^2-\frac{4\lambda }{11}-4$ is equal to

Answer 1

$\overrightarrow{AB}\perp \overrightarrow{PQ}$
$\Rightarrow \left[\right(4-\lambda )\hat{i}-4\hat{j}+\hat{k}]\cdot [\hat{i}+5\hat{j}-6\hat{k}]=0$
$\Rightarrow 4-\lambda -20-6=0$
$\Rightarrow \lambda =-22$

Now, $\frac{\lambda }{11}=-2$
$\therefore {\left(\frac{\lambda }{11}\right)}^2-\frac{4\lambda }{11}-4=4+8-4=8$